# Convergence in compact Hausdorff space

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• Apr 5th 2010, 09:56 PM
Arczi1984
Convergence in compact Hausdorff space
Hi,
I've problem with following convergence (in red shape). The second one term is tends to zero it is obvious but why the first one is tends to zero?
Thanks for any help.

http://img202.imageshack.us/img202/8928/lastkf.jpg
• Apr 5th 2010, 11:41 PM
Opalg
You are told that $x_n\to x$ in the sup norm. This means that, given $\varepsilon>0$, there exists N such that $n\geqslant N\ \Rightarrow\ \|x_n-x\|_{\text{sup}} = \sup\{|x_n(t) - x(t)|:t\in X\} <\varepsilon$. Take $t=p_n$ to see that $n\geqslant N\ \Rightarrow\ |x_n(p_n) - x(p_n)|<\varepsilon$. Thus $|x_n(p_n) - x(p_n)|\to0$ as $n\to\infty$.